Problem: Pluto's distance from the sun varies in a periodic way that can be modeled approximately by a trigonometric function. Pluto's maximum distance from the sun (aphelion) is $7.4$ billion kilometers. Its minimum distance from the sun (perihelion) is $4.4$ billion kilometers. Pluto last reached its perihelion in the year $1989$, and will next reach its perihelion in $2237$. Find the formula of the trigonometric function that models Pluto's distance $D$ from the sun (in billion $\text{km}$ ) $t$ years after $2000$. Define the function using radians. $ D(t) = $ How far will Pluto be from the sun in $2022$ ? Round your answer, if necessary, to two decimal places. $ $
Solution: Let's start by finding the distance to Pluto $u$ years after $1989$. Both sine and cosine can be used to model periodic contexts. We can decide which is better fitting by considering the $y$ -intercept. The sine function intercepts the $y$ -axis at its midline, and the cosine function intercepts the $y$ -axis at its peak. That way, we know Pluto achieved its minimum distance in $1989$, when $u = 0$. Since $\cos u$ reaches a peak at $0$, we'll use a cosine function to model this situation. Since we want it to achieve a minimum instead of a maximum at $u = 0$, we'll use a negative multiple of cosine. The amplitude of Pluto's distance is $\dfrac{7.4 - 4.4}{2} = 1.5$ billion $\text{km}$. The period of Pluto's distance is the amount of time between its minima: $2237 - 1989 = 248$ years. The midline of Pluto's distance is at the average of its perihelion and aphelion distance, or $\dfrac{7.4 + 4.4}{2} = 5.9$. Since the ordinary cosine function $f(u) = \cos u$ has period $2\pi$, midline $y = 0$, and amplitude $1$, we need to stretch it horizontally by a factor of $\dfrac{248}{2\pi}$, stretch it vertically by a factor of $1.5$, flip it vertically, and move it up $5.9$ units: $ D(u) = {-1.5}\cos\left({\dfrac{2\pi}{248}}u\right) + {5.9}$ Since $1989$ is $11$ years before $2000$, a year that's $t$ years after $2000$ will be $t + 11$ years after $1989$. So $u = t + 11$ : $ D(t) = {-1.5}\cos\left({\dfrac{2\pi}{248}}(t + 11)\right) + {5.9}$ Since $2022$ is $22$ years after $2000$, the distance from the sun to Pluto in $2022$ will be $\begin{aligned} D(22) &= {-1.5}\cos\left({\dfrac{2\pi}{248}}(22 + 11)\right) + {5.9} \\ &\approx 4.89 \end{aligned}$ A correct formula for $D(t)$ is: $ D(t) = -1.5\cos\left(\dfrac{\pi}{124}(t+11)\right) + 5.9$ Pluto's distance from the sun in $2022$ is: $ 4.89$ billion $\text{km}$